Induced Isometric Representations

TL;DR

This paper studies how to extend isometric representations from discrete to continuous parameters, preserving key properties, and applies this to classify a wide range of quantum dynamical systems with specific index values.

Contribution

It introduces a method to induce isometric representations from ext{N}^d to ext{R}_+^d, preserving index and irreducibility, and constructs a continuum of prime multiparameter CCR flows.

Findings

Induction preserves index and irreducibility for strongly pure isometric representations.

Constructs a continuum of prime multiparameter CCR flows with various indices.

Provides a classification framework for quantum dynamical semigroups.

Abstract

Let $\sigma$ be an isometric representation of $\mathbb{N}^d$ on a Hilbert space $\mathcal{H}$. We induce $\sigma$ to an isometric representation $V$ of $\mathbb{R}_{+}^{d}$ on another Hilbert space $\mathcal{K}$. We show that the map $\sigma \to V$, restricted to strongly pure isometric representations, preserves index and irreducibility. As an application, we show that, for $k \in \{0, 1,2,\cdots\} \cup \{\infty\}$, there is a continuum of prime multiparameter CCR flows (i.e, not a tensor product of two non-trivial $E_0$-semigroups) with index $k$.